The Franck-Hertz Experiment
Ryan Rubenzahl1, ∗ and Bo Peng1, † 1Department of Physics and Astronomy, University of Rochester, Rochester, New York, USA (Dated: December 6, 2017) We conduct the classic Franck-Hertz experiment in which electrons are accelerated through a mercury vapor and inelastically collide. We find the energy required to initiate inelastic collisions between electrons and mercury atoms to be 4.69 ± 0.05 eV, in agreement with the lowest excited 3 state of mercury (6 P0). The dependence of the Franck-Hertz curve on the tube temperature is explored, and we find a decrease in average peak spacing with increased temperature. We find the measurement of the lowest excited state to be stable with temperature if derived from minima spacings, and inconsistent if derived from the maxima spacings, in agreement with previous studies.
I. INTRODUCTION claim that the minima spacing actually grows with each successive minima. They model this effect by consider- James Franck and Gustav Ludwig Hertz, German sci- ing the following phenomena: Once an electron reaches entists at the University of Berlin, were the first to sufficient kinetic energy to undergo inelastic collisions demonstrate experimentally the quantized nature of mat- (4.67 eV), it will then on average travel one mean-free- ter in their famous experiment published in the German path before colliding with a Hg atom. In this time, the Physical Society on April 24th, 1914 [1]. They shared electron is further accelerated, gaining more energy and the Nobel Prize1 in 1925 for this “discovery of the laws thus possibly exciting one of the higher Hg energy levels 3 3 governing the impact of an electron upon an atom.” (6 P1 or 6 P2). When the voltage has been sufficiently The original experiment consisted of measuring the increased (roughly double the first minima), electrons current of electrons that were accelerated through a will be able to undergo two collisions before reaching the heated tube containing mercury (Hg) gas. Once the elec- grid, causing this effect of added energy to occur twice. trons reach a kinetic energy corresponding to the lowest The effect of this is a linear growth of the minima sepa- 3 ration ∆V with the minima order n: excited state of Hg (the 6 P0 state, see Fig 1), inelastic collisions between the electrons and Hg atoms will occur, ` and the Hg atom will be excited to this state. Electrons ∆V (n) = 1 + (2n − 1) Va, (1) L accelerated to a kinetic energy corresponding to the next 3 excited state (6 P1) will also excite the Hg atoms, how- where Va is the accelerating voltage at which inelastic ever this state is unstable, with a lifetime ∼ 105 times collisions begin to occur (4.67 V) and L is the distance 3 3 shorter than the 6 P0 state [2]. The 6 P1 state decays between the cathode and the grid. See Sec III of [3] for almost immediately back to the ground state by the spon- a derivation of Eq 1. The slope of this line allows the taneous emission of a photon, and is then ready to be ex- determination of the mean free path `, and the intercept 3 cited again. Hence while Hg atoms in the 6 P0 state will allows the determination of the lowest excitation energy collide elastically with another incoming electron, with Ea = eVa (e = electron charge). Since the electrons will the electron losing a negligible amount of kinetic energy follow a Maxwell-Boltzmann velocity distribution, their in the process (because of the large mass difference), in the same time period there will be ∼ 105 collisions ex- 3 citing other Hg atoms to the 6 P1 state. Franck and Hertz’s original experiment included a window through which the wavelength of the emitted photons could be measured, which they found to be 2536 A˚ [2]. By measuring the current of electrons exiting the tube 6P 3P2 of gas as a function of their energy, Franck and Hertz observed dips in the measured current corresponding to 3 the onset of inelastic collisions. While most analyses of 6P P1 the Franck-Hertz experiment assume the current minima 6P 3P0 3 occur at integer multiples of the 6 P1 excitation energy (4.89 eV, see Fig 1), Rapior, Sengstock, and Baev [3] 2536 Å 5.46 eV 4.89 eV 4.67 eV
6S 1S0 ∗ [email protected] † [email protected] FIG. 1: Energy levels of Hg relevant to this analysis 1 https://www.nobelprize.org/nobel_prizes/physics/ laureates/1925/ [2, 3]. 2
If an electron moving through the tube happens to have a kinetic energy corresponding to the an energy level of Hg, the electron may collide inelastically with a Hg pA atom, resulting in the electron no longer having enough energy to overcome the retarding potential. As a result of V these collisions, at a particular value of the accelerating voltage (corresponding to electrons with kinetic energy Vret matching that of the Hg excited state), a measurable + anode drop in the current recorded by the picoammeter can be seen. Continuously increasing the accelerating voltage grid will result in electrons again reaching the required kinetic Hg energy to inelastically collide with a Hg atom, causing a Hg Hg second drop in the measured current. Continuing this e-
+ process results in repeated dips in the measured current Hg Vacc at regular intervals (Fig 4), with the spacing between cathode minima corresponding to the energy required to excite filament the Hg atoms. This is the famous result of the original Franck-Hertz experiment, which Einstein, after hearing a presentation of the results by Franck at a conference, remarked to be “so lovely it makes you cry” [4]. + The density of the Hg gas present in the tube is con- trolled by the temperature of the oven enclosing the tube. In our analysis, we record measurements at five oven tem- Vfil peratures ranging between 135 ◦C and 175 ◦C in 10 de- gree intervals. At each temperature, a “run” consists FIG. 2: Schematic of the Franck-Hertz apparatus used. of recording the current measured by the picoammeter for accelerating voltages from 3 V to 40 V, in steps of around 0.1 V. Higher oven temperatures correspond to a mean free path ` is given by denser Hg vapor and therefore a smaller mean free path for the electrons, reducing the probability of an electron 1 kT kT navigating the thick Hg cloud. We therefore expect to ` = √ = √ = √ , (2) 2 measure smaller currents at higher temperatures, as well 2Nσ 2pσ 2p(πR0) as observe a decrease in the slope of the line from Eq 1. where R0 ≈ 1.5 A˚ is the cross-sectional radius of a Hg atom, and the Hg vapor density N is strongly sensitive to the temperature T of the Hg gas via the ideal gas law III. DATA COLLECTION p = NkT . Thus we expect the mean free path of the electrons, and hence the minima spacings, to vary with The data are recorded using the Franck-Hertz Lab- the oven temperature as well. Specifically, the average VIEW computer program. The LabVIEW program au- spacing between minima should decrease with increased tomates data collection after specifying a filament voltage temperature. In this analysis we consider the dependence and start, stop and step-size values to sweep the acceler- of the measured current minima on the oven temperature ating voltage. For all runs, we use a filament voltage of to explore this effect. Vfil = 5 V, a retarding voltage of Vret = 1.5 V, and sweep the accelerating voltage from 3 V to 40 V in steps of 0.1 V. This is done for oven temperatures of T = 135 ◦C, II. EXPERIMENTAL APPARATUS 145 ◦C , and 175 ◦C. For T = 155 ◦C and 165 ◦C, a step size of 0.05 V is used instead. The measurements for each Fig 2 shows a schematic of our experimental setup. run are displayed in Fig 3. Results for two different tube The filament voltage Vfil heats a filament resulting in temperatures are shown in Fig 4. the emission of electrons onto the cathode. These elec- Note that we can immediately see that the minima trons are then accelerated from the cathode to the grid locations are not equally spaced across different temper- by the electric field created by the accelerating voltage ature runs. Fig 4 shows the smoothed data for runs at ◦ ◦ Vacc. Electrons that reach the grid but have less energy T = 155 C and T = 175 C. From this it can be clearly than that corresponding to the retarding potential Vret seen that the average minima spacing shrinks with in- are halted and collected by the grid. Those with greater creased temperature, as expected. This suggests that energy are able to overcome the retarding potential and conclusions drawn directly from measurements of the av- make their way to the anode where they are measured as erage minima spacings are unreliable. a current by the Keithley picoammeter (pA in Fig 2). To estimate the uncertainty in our measurement of the 3
135 ∘ C 12.5
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0.0 10 20 30 40 10 20 30 40 145 ∘ C 2.5
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10∘ 20 30 40 10 20 30 40 0.8 155 C
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Anode Current [nA] AnodeCurrent 0.0 0.30 10 20 30 40 10 20 30 40 165 ∘ C 0.25
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0.00 10 20 30 40 10 20 30 40 0.125 175 ∘ C 0.100
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Accelerating Voltage [V]
FIG. 3: Data (left) collected for each temperature, and the smoothed and analyzed data (right). The vertical dashed lines correspond to bin edges, and the blue curves are the individual polynomials being fit in each bin. (•) points denote located minima, and (•) points denote located maxima. 4
80 0.7 155 ∘ C run 12 0.6 run 13 60 run 14 0.5 run 15 40 run 16 0.4
nA] 20